Модель визначення метричного тензора телекомунікаційної мережі на основі криволінійної системи координат

Abstract

Визначено метричний тензор, символи Крістофеля, тензори Рімана, Річі, скаляр кривизни простору для різних просторів. Наведено приклад визначення метричного тензору на основі теореми косинусів. Вперше визначено компоненту метричного тензора векторів з використанням теореми косинусів для чотирикутника з врахуванням двосто- роннього зв’язку між кожною парою вузлів. Запропоновано збільшити кількість компо- нент метричного тензора, що дасть змогу представити метрику у симетричному тензор- ному полі для опису деформації ріманової метрики, яку застосовують у потоках Річчі.

The tensor representation of telecommunication network parameters for various coordinate systems is described. The number of two-way links between nodes at the virtual level of the network is determined. A multidimensional coordinate system, the components of which may be various network parameters, such as the load between nodes, is considered. The state of the network is represented in a covariant and contravariant coordinate system. For assisted covariant differentiation described the possibility of taking into account changes in the state based on the Christoffel symbols. The definition of Riemann tensors is made. On the basis of the curvature tensor, the Ricci tensor is obtained by holding a convolution in a pair of indices, for example, in the first and third indices. In addition, another convolution was made on the Ricci tensor, which led to a scalar, which is called scalar curvature of space. The definition of the metric tensor for Euclidean and hyperbolic space is considered. To represent hyperbolic space, it is suggested to use a Poincare disk, which is a canonical unit disk. The description of the Möbius transformation, which is used to display the virtual coordinates on a Poincare disk, is given. An example of a metric tensor determination based on the cosine theorem is described for two cases:1) when there is a common point for two vectors; 2) when there is no such point. An increase in the number of components of a metric tensor has been proposed, which allows us to represent a metric in a symmetric tensor field for describing the deformation of the Riemann metric used in the Ricci flows. For a network of four nodes, for the first time, the component of the metric vectors tensor was determined using the cosine of the quadrangle, taking into account the two-way connection between each pair of nodes.The case where the load between the nodes is described by means of the exponential distribution law is considered. The component of the metric tensor for such a case and the differential of this component for the Ricci stream are determined.